Metamath Proof Explorer


Theorem nanbi2

Description: Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by Anthony Hart, 1-Sep-2011) (Proof shortened by SF, 2-Jan-2018)

Ref Expression
Assertion nanbi2
|- ( ( ph <-> ps ) -> ( ( ch -/\ ph ) <-> ( ch -/\ ps ) ) )

Proof

Step Hyp Ref Expression
1 nanbi1
 |-  ( ( ph <-> ps ) -> ( ( ph -/\ ch ) <-> ( ps -/\ ch ) ) )
2 nancom
 |-  ( ( ch -/\ ph ) <-> ( ph -/\ ch ) )
3 nancom
 |-  ( ( ch -/\ ps ) <-> ( ps -/\ ch ) )
4 1 2 3 3bitr4g
 |-  ( ( ph <-> ps ) -> ( ( ch -/\ ph ) <-> ( ch -/\ ps ) ) )